A particle starts from origin O from rest and moves with a uniform acceleration along the positive $$x$$-axis. Identify all figures that correctly represent the motion qualitatively. (a = acceleration, v = velocity, x = displacement, t = time)

Given that the particle starts from rest ($$u = 0$$) with uniform acceleration ($$a = \text{constant}$$):

At time $$t$$:

$$v = u + at \implies v = at$$

Since $$a$$ is a positive constant, the velocity-time ($$v-t$$) graph is a straight line passing through the origin with a positive slope.

For displacement:

$$x = ut + \frac{1}{2}at^2 \implies x = \frac{1}{2}at^2$$

The displacement-time ($$x-t$$) graph is a parabola opening upwards.

For acceleration:

$$a = \text{constant}$$

The acceleration-time ($$a-t$$) graph is a horizontal line.

Based on these relations:

• Graph (A): Correct (represents $$x \propto t^2$$)
• Graph (B): Correct (represents $$v \propto t$$)
• Graph (D): Correct (represents constant $$a$$)

Hence, graphs (A), (B), and (D) are correct.